When Dr. Yen Lo makes his little "yak dung" joke he parodies the famous advertising jingle "(Winston) tastes good like a cigarette should". This cigarette and its advertising slogan weren't introduced until 1954, a year after the end of the Korean War, so Yen Lo couldn't have made the joke. This error is copied from the novel.
After the fight between Marco and Chunjin, the movie marquee shows Pirates of Tortuga playing with Pinocchio. Although Pinocchio was released in 1940 and could be playing, Pirates of Tortuga wasn't released until 1961. The time frame in the movie is 2 years after the events in 1952, therefore 1954.
Senator Jordan tells Mrs. Iselin that he will begin "impeachment proceedings on the floor of the Senate" against Johnny Iselin if he tries for the vice-presidential nomination. Iselin similarly threatens to have Jordan impeached. First, impeachment proceedings begin in the House of Representatives. Second, Senators can't be impeached. They can be expelled by Senate vote but not impeached.
While speaking on the Senate floor, Iselin addresses the chair as "Mr. Speaker"; the presiding officer of the US Senate is the "President" (a post officially held by the Vice President), not the "speaker", thus the form of address is "Mr. President".
When Marco and Rosie are talking in the vestibule of the railroad car, she tells him, "I live on 64th Street, a few doors from The Modern Museum of Art". (She should correctly have said, "The Museum of Modern Art".)
Shaw's character wears the stripes of a Sergeant First Class on both his fatigue uniform in Korea and his dress uniform coat when returning to the U.S., although he is referred to in the film as "Staff Sergeant Raymond Shaw," which is actually one rank lower than the stripes he wears.
When Captain Marco is shown a photo of the Communist official Gomel at the birthday party, the first two views of the photo shows the little boy to Gomel's left and the girl to his right. As they cut away and come back for our third view, the boy is to the right and the little girl is on the left. The image has been reversed.
When Marco spots Shaw's Congressional Medal of Honor among the papers and debris on the floor. He reaches down and retrieves the medal from within the pile with his right hand but when the camera comes in for a close-up, the medal has suddenly switched to Marco's left hand.
When Marco and Rosie are talking on the train, the camera occasionally switches to close ups and we only see one head. Both are lit so a strong shadow is on one side of the head and a weaker one is on the other. In Marco's case the strong shadow is on the left as we look; which means Rosie's strong shadow should be on the right as we look, for she stands opposite Marco. However, the shadows are the same for both people, as if they stood on the same spot in relation to the lights' positions.
As Shaw arrives at the airport at the beginning of the film, several discrepancies on his uniform coat can be seen: he has two overseas bars at the bottoms of both sleeves (should be only on the left sleeve); he wears no three-year service stripes, even though he has obviously been in the service longer than three years; and he wears no ribbons for the National Defense Service Medal, the United Nations Service Medal, or for World War II service (his age and rank indicate he would also have served during that war).
Marco's dress uniform coat has two discrepancies: he wears no ribbon for the National Defense Service Medal, to which every Korean War veteran was entitled; and even though he has been assigned to Washington, D.C., for some time, he continues to wear his previous 24th Infantry Division patch on his left shoulder instead of the Military District of Washington patch.
During the scene on the train, you can see the countryside passing by outside the window. It's obvious that the actors are not really being filmed on a moving train, but the footage of the moving landscape was shot at an angle. So the camera that's aimed directly towards the window depicts a landscape that's bizarrely moving away from the train on an angle.
In Shaw's flashback of the summer with Josie Jordan, he is seen eating a meal with Josie and Senator Jordan. Halfway through the shot, it is simply reversed. You then see Shaw removing a piece of meat from his mouth with a fork and attaching it back to the meat on his plate with a knife.
The goof items below may give away important plot points.
The edition of the New York Post that announces the slaying of Senator Jordan and his daughter carries the following headline above the masthead: "Violent Hurricane Sweeps Midwest; 20 Dead, Hundreds Homeless". Hurricanes cannot reach the American Midwest, only its coastline. The Midwest has tornadoes.
At the end of the movie, when Raymond Shaw targets the presidential candidate Benjamin Arthur through the sniper scope, the first shot shows people sitting directly behind the Arthur. In subsequent shots, no one is sitting behind Arthur.
Just before Raymond shoots Senator Jordan, the Senator is holding a milk carton in front of his chest. Just as Raymond shoots him through the carton, he is holding it in front of his left arm. Being shot in the arm would not have caused him to fall unconscious. Also, the milk is shown only coming out of the front of the carton. It would have also been leaking from the back if the bullet had gone through it and hit the Senator.
The pistol that Raymond uses at the Jordan house is a Colt Official Police revolver, fitted with a silencer ("sound suppressor" is the correct term), but most revolvers can't be silenced/suppressed - the exhaust gases escape out the side. Only cartridge loaded pistols (in which the exhaust gases escape out the end of the barrel) can be silenced/suppressed. The only revolver that can be suppressed is the Russian Model 1895 Nagant, because of its unique design.
When Raymond assembles his rifle for the climactic assassination, the telescopic sight is removed from the case completely separate from the rest of the firearm. This would actually necessitate "sighting in" the weapon, by firing several test shots and adjusting the "scope" based on their results, before any degree of accuracy is possible.